Symmetric Factorial Designs in Blocks

  • R. A. BaileyEmail author


We consider factorial designs in blocks, where there are two treatment factors with the same number of levels, and both must be orthogonal to blocks. It is shown that these designs are duals of semi-Latin squares, and that the dual is optimal if and only if the semi-Latin square is optimal, for a wide range of optimality criteria. The optimal designs are described in language relevant for the factorial setting, which is shown to have applications in experiments on the interaction between humans and machines.


Dual design Factorial design Human-machine interaction Optimal design Semi-Latin square Trojan square 

AMS Subject Classification

62K05 62K10 62K15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bailey, R.A., 1988. Semi-Latin squares. Journal of Statistical Planning and Inference, 18, 299–312.MathSciNetCrossRefGoogle Scholar
  2. Bailey, R.A., 1992. Efficient semi-Latin squares. Statistica Sinica, 2, 413–437.MathSciNetzbMATHGoogle Scholar
  3. Bailey, R.A., 2008. Design of Comparative Experiments. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
  4. Bailey, R.A., Cameron, P.J., 2007. What is a design? How should we classify them? Designs, Codes and Cryptography, 44, 223–238.MathSciNetCrossRefGoogle Scholar
  5. Bailey, R.A., Royle, G., 1997. Optimal semi-Latin squares with side six and block size two. Proceedings of the Royal Society of London, Series A, 453, 1903–1914.MathSciNetCrossRefGoogle Scholar
  6. Bose, R.C., 1938. On the application of the properties of Galois fields to the problem of construction of hyper-Graeco-Latin squares. Sankhyā, 3, 323–338.Google Scholar
  7. Brickell, E.F., 1984. A few results in message authentication. Congressus Numerantium, 43, 141–154.MathSciNetzbMATHGoogle Scholar
  8. Cheng, C.-S., 1978. Optimality of certain asymmetrical experimental designs. Annals of Statistics, 6, 1239–1261.MathSciNetCrossRefGoogle Scholar
  9. Cheng, C.-S., Bailey, R.A., 1991. Optimality of some two-associate-class partially balanced incomplete-block designs. Annals of Statistics, 19, 1667–1671.MathSciNetCrossRefGoogle Scholar
  10. Chigbu, P.E., 2003. The “best” of the three optimal (4×4)/4 semi-Latin squares. Sankhyaā, 65, 641–648.MathSciNetzbMATHGoogle Scholar
  11. Darby, L.A., Gilbert, N., 1958. The Trojan square. Euphytica, 7, 183–188.CrossRefGoogle Scholar
  12. Dean, A.M., Lewis, S.M., 1983. Upper bounds for average efficiency factors of two-factor interactions. Journal of the Royal Statistical Society, Series B, 45, 252–257.zbMATHGoogle Scholar
  13. Fisher, R.A., 1942. The theory of confounding in factorial experiments in relation to the theory of groups. Annals of Eugenics, 11, 341–353.MathSciNetCrossRefGoogle Scholar
  14. Fisher, R.A., 1945. A system of confounding for factors with more than two alternatives, giving completely orthogonal cubes and higher powers. Annals of Eugenics, 12, 282–290.MathSciNetzbMATHGoogle Scholar
  15. Healey, P.G.T., Swoboda, N., Umata, I., Katagiri, Y., 2002. Graphical representation in graphical dialogue. International Journal of Human Computer Studies, 57, 373–395.Google Scholar
  16. James, A.T., Wilkinson, G.N., 1971. Factorization of the residual operator and canonical decomposition of nonorthogonal factors in the analysis of variance. Biometrika, 58, 279–294.MathSciNetCrossRefGoogle Scholar
  17. Jarrett, R.G., 1993. On the construction and properties of row-column designs using their duals. Preprint, University of Adelaide.Google Scholar
  18. John, J.A., Mitchell, T.J., 1977. Optimal incomplete block designs. Journal of the Royal Statistical Society, Series B, 39, 39–43.MathSciNetzbMATHGoogle Scholar
  19. John, J.A., Smith, T.M.F., 1972. Two factor experiments in non-orthogonal designs. Journal of the Royal Statistical Society, Series B, 34, 401–409.MathSciNetzbMATHGoogle Scholar
  20. Nelder, J.A., 1965. The analysis of randomized experiments with orthogonal block structure. II. Treatment structure and the general analysis of variance. Proceedings of the Royal Society of London, Series A, 283, 163–178.MathSciNetCrossRefGoogle Scholar
  21. Patterson, H.D., Williams, E.R., 1976. Some theoretical results on general block designs. Congressus Numerantium, 15, 489–496.MathSciNetzbMATHGoogle Scholar
  22. Pearce, S. C., 1968. The mean efficiency of equi-replicate designs. Biometrika, 55, 251–253.MathSciNetzbMATHGoogle Scholar
  23. Phillips, N.C.K., Wallis, W.D., 1996. All solutions to a tournament problem. Congressus Numerantium, 114, 193–196.MathSciNetzbMATHGoogle Scholar
  24. Preece, D.A., Freeman, G.H., 1983. Semi-Latin squares and related designs. Journal of the Royal Statistical Society, Series B, 45, 267–277.zbMATHGoogle Scholar
  25. Shah, K.R., Sinha, B.K., 1989. Theory of Optimal Designs. Springer, New York.CrossRefGoogle Scholar
  26. Soicher, L.H., 1999. On the structure and classification of SOMAs: generalizations of mutually orthogonal Latin squares. Electronic Journal of Combinatorics, 6, R32. 15 pages, Available at: MathSciNetzbMATHGoogle Scholar
  27. Soicher, L.H., 2000. SOMA Update. Scholar
  28. Suen, C.-Y., Chakravarti, I.M., 1986. Efficient two-factor balanced designs. Journal of the Royal Statistical Society, Series B, 48, 107–114.MathSciNetzbMATHGoogle Scholar
  29. Yates, F., 1935. Complex experiments (with discussion). Journal of the Royal Statistical Society, Supplement, 2, 181–247.CrossRefGoogle Scholar
  30. Yates, F., 1939. The comparative advantages of systematic and randomized arrangements in the design of agricultural and biological experiments. Biometrika, 30, 440–466.CrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2011

Authors and Affiliations

  1. 1.School of Mathematical Sciences, Queen MaryUniversity of LondonLondonUK

Personalised recommendations